Question

Which of the following could be true if \(f^{\prime \prime}(x)=x^{-1 / 3} ?\) (a) \(f(x)=\frac{3}{2} x^{2 / 3}-3 \quad\) (b) \(f(x)=\frac{9}{10} x^{5 / 3}-7\) \((\mathbf{c}) f^{\prime \prime \prime}(x)=-\frac{1}{3} x^{-4 / 3} \quad\) (d) \(f^{\prime}(x)=\frac{3}{2} x^{2 / 3}+6\)

Short answer

Therefore, the answer is (c) \(f''(x) = -\frac{1}{3} x^{-4 / 3}\).

Step-by-step solution

Obtain the first derivative

Firstly, integrate \(f^{\prime \prime}(x)=x^{-1 / 3}\) to find \(f^{\prime}(x)\). The integral of \(x^{-1 / 3}\) is \(\frac{3}{2} x^{2 / 3} + C_1\), where \(C_1\) is the constant of integration.

Obtain the original function

To find \(f(x)\), perform a second integration on \(f^{\prime}(x)=\frac{3}{2} x^{2 / 3} + C_1\). The integral of \(\frac{3}{2} x^{2 / 3}\) is \(\frac{9}{5} x^{5 / 3}\) and the integral of \(C_1\) is \(C_1 x\). Therefore, \(f(x)=\frac{9}{5} x^{5 / 3}+C_1 x + C_2\), where \(C_2\) is another constant of integration.

Compare with options

Now, compare function \(f(x)=\frac{9}{5} x^{5 / 3}+C_1 x + C_2\) with choices (a) and (b). It doesn't match with either (a) or (b). Check also choice (d). The function that is compared with choice (d) is the first derivative \(f^{\prime}(x) = \frac{3}{2} x^{2 / 3} + C_1\), which is in the same form but doesn't exactly meet the choice. This leaves us with choice (c), f'''. For that, directly take the derivative of \(f^{\prime \prime}(x)=x^{-1 / 3}\) without needing to use constants. The derivative of \(x^{-1 / 3}\) results in \(-\frac{1}{3} x^{-4 / 3}\), which matches with choice (c) perfectly.